Equations ()
and () describe the hydrodynamic
behavior of an electrically charged, incompressible fluid under the
action of an external force \(\mathbf {f}\)  and the electromagnetic force
\(\mathbf {j}\times \mathbf {B}\)  multiplied by the Hartmann number, which acts as a
coupling coefficient.
The electromagnetic force is self-consistently generated by the electromagnetic fields \(\mathbf {E}\)  and
\(\mathbf {B}\)  satisfying equations (),
(), and
().
These last three equations describe the electromagnetic submodel in
the magneto-hydrodynamics approximation [1]}.
The incompressibility of the fluid velocity and the solenoidal nature
of the magnetic field require that both \(\mathbf {u}_0\)  and \(\mathbf {B}_0\) 
in (REF ) are divergence free.
